无解或所有的实数

As with equalities, it's possible for inequalities to have no solution. And, just to make your life a little more fun, it's also possible that the solution to an inequality can include all real numbers.

Here are a couple graphics to get you thinking before looking at some more complex examples.

Inequalities with no solution

What you're looking for in this case is nonsense. Yes, that's right: you are looking for nonsense. If you come across an inequality that indicates information that is not true, then the inequality has no solution.

Let's look at a few examples.

There is no way that an absolute value can be less than or equal to a negative number. By definition, absolute values are greater than or equal to $0$ (i.e., $0$ or positive integers). This inequality violates that definition, making it nonsense. We can say that this inequality has no solution.

Keep in mind that sometimes you may need to isolate the absolute value in order to solve for it.

When we simplify this inequality to get the $x$ terms on one side, here's what we get:

$$\begin{align} 5x - 9 &> 3x + 4 + 2x\\ 5x - 9 &> 5x + 4\\ -9 &> 4 \end{align}$$

$-9$ is definitely not greater than $4$. This is also nonsense, which means that the inequality has no solution.

Nope, this isn't true either. This inequality violates another definition: no squared values can be negative unless they are multiplied by negative numbers. This is also nonsense, which means that this inequality has no solution.

There is one more possibility of when inequality have no solutions. Look at the example below. The shaded regions never touch, so there is no solution. This is known as an OR compound inequality.

Inequalities with "all real numbers" solutions

In this situation, we are looking for inequalities that give us information that is ALWAYS true. Let's look at an example:

$$x + 7 > x + 3$$

This statement will always be true. It doesn't matter if $x$ is a positive or negative number. Adding $7$ will always produce a greater number than adding $3$. Because this is always true, we can say that this inequality's solution is all real numbers.

We can write it in interval notation as: $(-\infty,\infty)$. We can't fully reach infinity, so we have to use an exclusive interval. You could further simplify this inequality by putting the $x$ terms on one side. Let's try that and see if our answer changes:

$$\begin{align} x + 7 &> x + 3\\ (x + 7) - 7 &> (x + 3) - 7\\ x &> x - 4\\ x - x &> (x - 4) - x\\\ \\ 0 &> -4 \end{align}$$

Nope! Our answer stays the same because this expression is always true. $0$ will always be greater than $-4$. This confirms that the inequality's solution is all real numbers.

Inequalities with no solution or all real numbers

Sometimes, it's possible to solve an inequality and realize it is meaningless! Let me give a couple of examples:

$$3x + 3 < 4x < 3x -3$$

We can work through this and get:

$$3x + 3 < 4x < 3x -3\\ (3x + 3) -3x < (4x) - 3x < (3x -3) - 3x\\\ \\ 3 < x < -3$$

But we can't pick a value that is greater than $3$ and less than $-3$, so this inequality has no solution! When solving an inequality, keep your eyes open for such scenarios where the answer you get makes no sense.

Remember, $x^2$ is always nonnegative (in other words, $0$ or a positive number), so it can never be less than zero!

In this case, all values of $x$ satisfy the inequality. Any number squared will be $0$ or greater. We say $x$ has the solution set "all real numbers" when this is the case.

It is possible to encounter certain inequalities where there is no solution, or where everything is a solution.

(Video) Solve No Solution and All Real Number Inequalities

by Monica Wieneke

This video by Monica Wieneke shows you some inequalities with no solution or infinite solutions.

$\large{\text{Summary}}$

You can get inequalities that are false for any variable. These inequalities have no solution. You can also get inequalities that are true for any variable. These have an infinite number of solutions.

The first example she does is: $7p-11p+3\geq3-4p$. $$\begin{align} 7p-11p+3&\geq3-4p\\ -4p+3&\ge3-4p\\ -4p{\color{#8925AE}{+4p}} +3&\ge3-4p{\color{#8925AE}{+4p}}\\ 3&\ge3 \end{align}$$

Here we get a true statement. $3$ is greater than or equal to $3$. So we know this inequality's solution is all real numbers or infinite solutions. No matter what number you put in for $p$, you get a true statement.

The next example she does is: $2(3d-4)<4+6d-15$. $$\begin{align} 2(3d-4)&<4+6d-15\\ 6d-8&<6d-11\\ 6d{\color{#8925AE}{-6d}}-8&<6d{\color{#8925AE}{-6d}}\color{black}-11\\ -8&<-11 \end{align}$$ Here we get a false statement because $-8$ is not less than $-11$. This means the inequality has no solutions.

Try it!

Try the next two problems on your own and see if they have no solution or all real numbers as a solution.

$$\begin{align} 4(3-2x) &> 2(6-4x)\\ 12 - 8x &> 12 - 8x\\ 12 &> 12 \\ \end{align}$$

No matter what we plug in for $x$, we can always simplify out to $12 < 12$. Since that's not true ($12$ can't ever be less than itself), there are no solutions.

$$\begin{align} -4n-25 &< -4(n+6)\\ -4n - 25 &< -4n - 24\\ - 25 &< -24 \end{align}$$

You'll need the distributive property to simplify inequalities.

No matter what variable you plug in for $n$, you can always simplify the equation and get $-25 < -24$. Since we know $- 25$ is less than $-24$, the inequality will always be true. So, this inequality has infinite solutions.